Question

The proposed mechanism for the formation of hydrogen bromide can be written in a simplified form as: \(\begin{array}{ll}\operatorname{Br}_{2}(g) \stackrel{k_{1}}{k_{1}} 2 \operatorname{Br}(g) & \text { Fast } \\\ \operatorname{Br}(g)+\mathrm{H}_{2}(g) \stackrel{k_{2}}{\longrightarrow} \operatorname{HBr}(g)+\mathrm{H}(g) & \text { Slow } \\\ \mathrm{H}(g)+\mathrm{Br}_{2}(g) \stackrel{k_{3}}{\longrightarrow} \mathrm{HBr}(g)+\operatorname{Br}(g) & \text { Fast }\end{array}\)

Short answer

The simplified rate law for the overall formation of hydrogen bromide is rate = \(k'[Br_2]^{1/2}[H_2]\), where \(k'\) is a combined rate constant from the mechanism.

Step-by-step solution

Determine Rate-Determining Step

The rate-determining step is the slowest step in a reaction mechanism and controls the overall rate of the reaction. According to the given mechanism, the second step is the slow step, which means it is the rate-determining step.

Write Rate Law for Rate-Determining Step

The rate law can be written based on the rate-determining step. The rate law only includes the reactants in the slow step and their stoichiometries as exponents in the rate equation. For the given mechanism, the rate law for the second step is rate = \(k_2[Br][H_2]\).

Identify Intermediate Species

In a mechanism, intermediate species are formed in one step and consumed in another. Since \(Br\) atoms are produced in the first step and consumed in the second and third steps, they are intermediates. Intermediates should not appear in the final rate law for the overall reaction.

Express the Intermediate Concentration in Terms of Known Quantities

Since intermediates cannot appear in the final rate law, we must express their concentration in terms of known quantities. The concentration of \(Br\) is determined by the fast pre-equilibrium of the first step, where \(Br_2\) dissociates into 2 \(Br\) atoms. At equilibrium, \(rate_{forward} = rate_{reverse}\), which leads to \(k_1[Br_2] = k_{-1}[Br]^2\). Solving for \([Br]\), we get \([Br] = \sqrt{\frac{k_1}{k_{-1}}[Br_2]}\).

Substitute Intermediate Concentration into Rate Law

Substituting the expression for \([Br]\) into the rate law for the rate-determining step gives the rate law in terms of the original reactants only. The final rate law becomes rate = \(k_2\left(\sqrt{\frac{k_1}{k_{-1}}[Br_2]}\right)[H_2]\).

Simplify the Rate Law

The expression inside the square root \(\sqrt{\frac{k_1}{k_{-1}}}\) can be combined into a single rate constant, \(k'\), as it only involves constants and does not change with the concentration of the reactants. The simplified rate law is rate = \(k'[Br_2]^{1/2}[H_2]\) where \(k' = k_2\sqrt{\frac{k_1}{k_{-1}}}\).

Key concepts

Rate-Determining Step

When studying chemical reactions, understanding the rate-determining step is essential. This is the slowest step in a sequence of reactions that make up the complete reaction mechanism, effectively dictating the speed at which the overall reaction proceeds. Think of it as the bottleneck of a process: no matter how fast the other steps are, the overall rate is limited by this slowest stage.

In our example, the formation of hydrogen bromide involves several steps, but it is the second step that is labeled as 'Slow,' indicating that it is the rate-determining step. Identifying this step helps us focus on the significant elements that affect the reaction rate, allowing us to formulate an accurate rate law.

Rate Law

The rate law is a mathematical expression that describes the relationship between the rate of a chemical reaction and the concentration of its reactants. To construct a rate law, we use coefficients from the balanced equation to determine the reaction order for each reactant, which tells us how the rate of reaction depends on the reactant's concentration.

In the context of our exercise, the rate law for the rate-determining step is \( rate = k_2[Br][H_2] \). This equation implies that the rate of hydrogen bromide formation depends on the concentration of bromine atoms and hydrogen gas. The rate constant (\( k_2 \) in this case) captures all other variables that affect rate, including temperature and the presence of catalysts.

Reaction Intermediates

Reaction intermediates are transient species that appear in the course of a reaction mechanism but do not appear in the net equation for the overall reaction. They're like 'behind-the-scenes' actors—vital to the storyline but not seen in the final scene.

During the formation of hydrogen bromide, bromine atoms (\(Br\)) play the role of an intermediate, being produced in the first step and consumed in subsequent steps. Because they are not present in the reactants or products of the overall reaction, they are not included in the final rate law expression. However, their concentration is still critical to the reaction’s progress, thus it must be expressed in terms of reactants using the established equilibrium condition from the initial fast step.